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<h1>Maximize stopband attenuation of a lowpass IIR filter</h1>
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<pre class="codeinput">
<span class="comment">% "Linear Programming Design of IIR Digital Filters with Arbitrary</span>
<span class="comment">%  Magnitude Functions" by L.R. Rabiner, N.Y. Graham, and H.D. Helms</span>
<span class="comment">% (figures are generated)</span>
<span class="comment">%</span>
<span class="comment">% Designs a lowpass IIR filter using spectral factorization method where we:</span>
<span class="comment">% - minimize maximum stopband attenuation</span>
<span class="comment">% - have a constraint on the maximum passband ripple</span>
<span class="comment">%</span>
<span class="comment">%   minimize   max |H(w)|                      for w in the stopband</span>
<span class="comment">%       s.t.   1/delta &lt;= |H(w)| &lt;= delta      for w in the passband</span>
<span class="comment">%</span>
<span class="comment">% where we now have a rational frequency response function:</span>
<span class="comment">%</span>
<span class="comment">%   H(w) = sum_{m=0}^{M-1} b_m exp{-jwm} / sum_{n=1}^{N-1} a_n exp{-jwn}</span>
<span class="comment">%</span>
<span class="comment">% We change variables via spectral factorization method and get:</span>
<span class="comment">%</span>
<span class="comment">%   minimize   max R(w)                        for w in the stopband</span>
<span class="comment">%       s.t.   (1/delta)^2 &lt;= R(w) &lt;= delta^2  for w in the passband</span>
<span class="comment">%              R(w) &gt;= 0                       for all w</span>
<span class="comment">%</span>
<span class="comment">% where R(w) is the squared magnited of the frequency response</span>
<span class="comment">% (and the Fourier transform of the autocorrelation coefficients r).</span>
<span class="comment">% We represent R(w) = N_hat(w)/D_hat(w), where now R(w) is a rational</span>
<span class="comment">% function since we deal with IIR filter (see the reference paper).</span>
<span class="comment">%</span>
<span class="comment">% Variables are coeffients of the numerator, denoted as c, and</span>
<span class="comment">% denominator, denoted as d. delta is the allowed passband ripple.</span>
<span class="comment">% This is a quasiconvex problem and can be solved using bisection.</span>
<span class="comment">%</span>
<span class="comment">% Written for CVX by Almir Mutapcic 02/02/06</span>

<span class="comment">%*********************************************************************</span>
<span class="comment">% user's filter specs (for a low-pass filter example)</span>
<span class="comment">%*********************************************************************</span>
<span class="comment">% number of coefficients for the IIR filter (including the zeroth one)</span>
<span class="comment">% (also without loss of generality we can assume that d_0 = 1, which</span>
<span class="comment">%  is the zeroth coefficient of the autocorrelation denominator)</span>
M = 4; <span class="comment">% nominator</span>
N = 4; <span class="comment">% denominator</span>

wpass = 0.12*pi;   <span class="comment">% end of the passband</span>
wstop = 0.24*pi;   <span class="comment">% start of the stopband</span>
delta = 1;         <span class="comment">% maximum passband ripple in dB (+/- around 0 dB)</span>

<span class="comment">%*********************************************************************</span>
<span class="comment">% create optimization parameters</span>
<span class="comment">%*********************************************************************</span>
<span class="comment">% rule-of-thumb discretization (from Cheney's Approx. Theory book)</span>
sample_order = 30;
m = 15*(sample_order);
w = linspace(0,pi,m)'; <span class="comment">% omega</span>

<span class="comment">% A's are matrices used to compute the power spectrum</span>
Anum = [ones(m,1) 2*cos(kron(w,[1:M-1]))];
Aden = [ones(m,1) 2*cos(kron(w,[1:N-1]))];

<span class="comment">% passband 0 &lt;= w &lt;= w_pass</span>
ind = find((0 &lt;= w) &amp; (w &lt;= wpass));    <span class="comment">% passband</span>
Ap_num  = Anum(ind,:);
Ap_den  = Aden(ind,:);

<span class="comment">% transition band is not constrained (w_pass &lt;= w &lt;= w_stop)</span>

<span class="comment">% stopband (w_stop &lt;= w)</span>
ind = find((wstop &lt;= w) &amp; (w &lt;= pi));   <span class="comment">% stopband</span>
As_num  = Anum(ind,:);
As_den  = Aden(ind,:);

<span class="comment">%********************************************************************</span>
<span class="comment">% optimization</span>
<span class="comment">%********************************************************************</span>

<span class="comment">% use bisection (on the log of vars) to solve for the min stopband atten</span>
Us_top = 1e-0; <span class="comment">% 0 dB</span>
Us_bot = 1e-6; <span class="comment">% -60 dB (in original variables)</span>

<span class="keyword">while</span>( 20*log10(Us_top/Us_bot) &gt; 1)
  <span class="comment">% try to find a feasible design for given specs</span>
  Us_cur = sqrt(Us_top*Us_bot);

  <span class="comment">% formulate and solve the magnitude design problem</span>
  cvx_begin <span class="string">quiet</span>
    variable <span class="string">c(M,1)</span>
    variable <span class="string">d(N-1,1)</span>

    <span class="comment">% feasibility problem</span>
    <span class="comment">% passband constraints</span>
    (Ap_num*c) &lt;= (10^(+delta/20))^2*(Ap_den*[1;d]); <span class="comment">% upper constr</span>
    (Ap_num*c) &gt;= (10^(-delta/20))^2*(Ap_den*[1;d]); <span class="comment">% lower constr</span>
    <span class="comment">% stopband constraint</span>
    (As_num*c) &lt;= (Us_cur)*(As_den*[1;d]); <span class="comment">% upper constr</span>
    <span class="comment">% nonnegative-real constraint</span>
    Anum*c &gt;= 0;
    Aden*[1;d] &gt;= 0;
  cvx_end

  <span class="comment">% bisection</span>
  <span class="keyword">if</span> ~any(isnan(c)) <span class="comment">% feasible</span>
    fprintf(1,<span class="string">'Problem is feasible for stopband atten = %3.2f dB\n'</span>, <span class="keyword">...</span>
               10*log10(Us_cur));
    Us_top = Us_cur;
    b = spectral_fact(c);
    a = spectral_fact([1;d]);
  <span class="keyword">else</span> <span class="comment">% not feasible</span>
    fprintf(1,<span class="string">'Problem not feasible for stopband atten = %3.2f dB\n'</span>, <span class="keyword">...</span>
               10*log10(Us_cur));
    Us_bot = Us_cur;
  <span class="keyword">end</span>
<span class="keyword">end</span>

<span class="comment">% display the max attenuation in the stopband (convert to original vars)</span>
fprintf(1,<span class="string">'\nOptimum min stopband atten is between %3.2f and %3.2f dB.\n'</span>,<span class="keyword">...</span>
        10*log10(Us_bot),10*log10(Us_top));
disp(<span class="string">'Optimal IIR filter coefficients are: '</span>)
disp(<span class="string">'Numerator: '</span>), b
disp(<span class="string">'Denominator: '</span>), a

<span class="comment">%*********************************************************************</span>
<span class="comment">% plotting routines</span>
<span class="comment">%*********************************************************************</span>
<span class="comment">% frequency response of the designed filter, where j = sqrt(-1)</span>
H = ([exp(-j*kron(w,[0:M-1]))]*b)./([exp(-j*kron(w,[0:N-1]))]*a);

<span class="comment">% magnitude plot</span>
figure(1)
subplot(2,1,1)
plot(w,20*log10(abs(H)), <span class="keyword">...</span>
     [0 wpass],[delta delta],<span class="string">'r--'</span>, <span class="keyword">...</span>
     [0 wpass],[-delta -delta],<span class="string">'r--'</span>, <span class="keyword">...</span>
     [wstop pi],[10*log10(Us_top) 10*log10(Us_top)],<span class="string">'r--'</span>)
xlabel(<span class="string">'w'</span>)
ylabel(<span class="string">'mag H(w) in dB'</span>)
axis([0 pi -55 10]);

<span class="comment">% phase plot</span>
subplot(2,1,2)
plot(w,angle(H))
axis([0,pi,-pi,pi])
xlabel(<span class="string">'w'</span>), ylabel(<span class="string">'phase H(w)'</span>)
</pre>
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<pre class="codeoutput">
Problem is feasible for stopband atten = -30.00 dB
Problem not feasible for stopband atten = -45.00 dB
Problem is feasible for stopband atten = -37.50 dB
Problem not feasible for stopband atten = -41.25 dB
Problem not feasible for stopband atten = -39.38 dB
Problem is feasible for stopband atten = -38.44 dB
Problem is feasible for stopband atten = -38.91 dB

Optimum min stopband atten is between -39.38 and -38.91 dB.
Optimal IIR filter coefficients are: 
Numerator: 

b =

    0.0036
   -0.0011
   -0.0011
    0.0035

Denominator: 

a =

    0.2672
   -0.6977
    0.6343
   -0.1990

</pre>
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